
which is consistence with the \(f(\mathcal {G},\mathbf {T}^{2})\) 
equations of motion. It is notable that the condition \(\mathbb {U}=0\) , or
\(B=1\)  provides a non-dynamical model. Moreover the constraint of
geodesic fluid presents the FRW model [1]}.
Now, imposing the shear-free condition (i.e., \(\sigma =0\) ) along with
the constraint \(Y_{TF}=0\)  in Eq.(REF ), we have
\(a^{2}+\frac{1}{B}\left(a^{\prime }+a\frac{C^{\prime }}{C}\right)+\frac{1}{2}\left(\frac{\varphi _{11}}{B^{2}}-\frac{\varphi _{22}}{C^{2}}+\frac{\phi _{\mathbf {2}}}{2}\right)=0\) 
